Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 809689, 12 pages
doi:10.1155/2012/809689
Research Article
Hermite-Hadamard and Simpson-Like
Type Inequalities for Differentiable (α, m)-Convex
Mappings
Jaekeun Park
Department of Mathematics, Hanseo University, Chungnam-do, Seosan-si 356-706, Republic of Korea
Correspondence should be addressed to Jaekeun Park, jkpark@hanseo.ac.kr
Received 13 July 2011; Revised 13 November 2011; Accepted 28 November 2011
Academic Editor: Jewgeni Dshalalow
Copyright q 2012 Jaekeun Park. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The author establish several Hermite-Hadamard and Simpson-like type inequalities for mappings
whose first derivative in absolute value aroused to the qth q ≥ 1 power are α, m-convex. Some
applications to special means of positive real numbers are also given.
1. Introduction
Recall that, for some fixed α ∈ 0, 1 and m ∈ 0, 1, a mapping f : I ⊆ 0, ∞ → R is said to
be α, m-convex on an interval I if the inequality
f tx m1 − ty ≤ tα fx m1 − tα f y
1.1
holds for x, y ∈ I, and t ∈ 0, 1.
α
I the set of all α, m-convex mappings on I. For recent results and
Denote by Km
generalizations concerning m-convex and α, m-convex mappings, see 1–4.
α
I, let us denote
For the simplicities of notations, for f ∈ Km
mb
a mb
1
1
fa r − 2f
fmb −
fxdx.
a − mbSba f α, m, r r
2
mb − a a
1.2
2
International Journal of Mathematics and Mathematical Sciences
In 1, 3, Klaričić Bakula and Özdemir et al., proved the following Hadamard’s
inequalities for mappings whose second derivative in absolute value aroused to the q-th
q ≥ 1 power are α, m-convex.
Theorem 1.1. Let f : I ⊆ 0, b∗ → R be a twice differentiable mapping on the interior I0 of an
interval I such that f ∈ L1 a, b, where a, b ∈ I with a < b and b∗ > 0. If |f |q is α, m-convex
on a, b for α, m ∈ 0, 12 and q ≥ 1 with 1/p 1/q 1, then the following inequality holds:
mb − a 1 1/p q
q 1/q
,
μf a mνf b
a Sba f α, m, 2 ≤
2
6
1.3
where
α2 5α
1
,
,
ν
6α 2α 3
α 2α 3
mb − a 1 1/p q
q 1/q
,
μ0 f a mν0 f b
b Sba f α, m, 6 ≤
2
2
μ
1.4
1.5
where
Γα 2Γ q
q
μ0 ,
qα2 Γ qα1
Γα 2Γ q
q
1
ν0 ,
−
qα2 Γ qα1
q1 q2
1.6
where
Γx ∞
e−t tx−1 dt,
x > 0.
1.7
0
Theorem 1.2. Under the same notations in Theorem 2.2, if |f |q is α, m-convex on a, b for
α, m ∈ 0, 12 and q > 1 with 1/p 1/q 1, then the following inequality holds:
mb − a
b
Sa f α, m, 2 ≤
8
1/p q α 1/q
q 1
Γ p1
f a
m f b
. 1.8
α1
α1
Γ p 3/2
Note that for α, m ∈ {0, 0, α, 0, 1, 0, 1, m, 1, 1, α, 1} one obtains the following
classes of functions: increasing, α-starshaped, starshaped, m-convex, convex, and α-convex.
For the definitions and elementary properties of these classes, see 4–8.
For recent years, many authors present some new results about Simpson’s inequality
for α, m-convex mappings and have established error estimations for the Simpson’s
inequality: for refinements, counterparts, generalizations, and new Simpson’s type inequalities, see 1–3, 6.
In 9, Dragomir et al. proved the following theorem.
International Journal of Mathematics and Mathematical Sciences
3
Theorem 1.3. Let f : I ⊂ 0, ∞ → R be an absolutely continuous mapping on a, b such that
f ∈ Lp a, b, where a, b ∈ I with a < b. Then the following inequality holds:
b − a−1/p
b
Sa f 1, 1, 6 ≤
6
2q1 1
3 q1
1/q
f .
p
1.9
The readers can estimate the error f in the generalized Simpson’s formula without
going through its higher derivatives which may not exist, not be bounded, or may be hard to
find.
In this paper, the author establishes some generalizations of Hermite-Hadamard and
Simpson-like type inequalities based on differentiable α, m-convex mappings by using the
following new identity in Lemma 2.1 and by using these results, obtain some applications to
special means of positive real numbers.
α
2. Generalizations of Simpson-Like Type Inequalities on Km
I
In order to generalize the classical Simpson-like type inequalities and prove them, we need
the following lemma 6.
Lemma 2.1. Let f : I ⊆ 0, b∗ → R be a differentiable mapping on the interior I0 of an interval I,
where a, b ∈ I with 0 ≤ a < b and b∗ > 0. If f ∈ L1 a, b, then, for r ≥ 2 and h ∈ 0, 1 with
1/r ≤ h ≤ r − 1/r, the following equality holds:
Sba f α, m, r 1
pr, tf ta m1 − tbdt
2.1
1
t ∈ 0, ,
2
1
t∈
,1 .
2
2.2
0
α
a, b and each t ∈ 0, 1, where
for f ∈ Km
pr, t ⎧
1
⎪
⎪
t−
⎪
⎨
r
⎪
⎪
⎪
⎩t − r − 1
r
By the similar way as Theorems 1.1–1.3, we obtain the following theorems.
Theorem 2.2. Let f : I ⊂ 0, b∗ → R be a differentiable mapping on I0 such that f ∈ La, b,
α
a, b, for some α, m ∈ 0, 12 and
where a, b ∈ I with 0 ≤ a < b < ∞ and b∗ > 0. If |f | ∈ Km
mb > a, then, for any r ≥ 2, the following inequality holds:
b
Sa f α, m, r ≤ μ11 μ21 μ31 μ41 f a ν11 ν12 ν13 ν14 mf b,
2.3
4
International Journal of Mathematics and Mathematical Sciences
where
μ11 1
,
α 1α 2r α2
μ21 1
α 1r − 2α 2
,
α2
α2
2 α 1α 2r
α 1α 2r
μ31 2α 2 − 3r
r − 1α2
α2
,
α2
2 α 1α 2r
α 1α 2r
μ41 r − 1α2
α 2 − r
,
α2
α 1α 2r
α 1α 2r
2.4
1
ν11 2 − μ11 ,
2r
ν12 r − 22
− μ21 ,
8r 2
r − 22
− μ31 ,
8r 2
1
ν14 2 − μ41 .
2r
ν13 Proof. From Lemma 2.1 and using the properties of the modulus, we have the following:
1/r 1
b
− t f ta m1 − tbdt
Sa f α, m, r ≤
r
0
1/2 1 f ta m1 − tbdt
t−
r
1/r
r−1/r r−1
− t f ta m1 − tbdt
r
1/2
1
r − 1 f ta m1 − tbdt.
t−
r
r−1/r
2.5
Since |f | is α, m-convex on a, b, we know that for any t ∈ 0, 1
f ta m1 − tb ≤ tα f a m1 − tα f b.
By 2.5 and 2.6, we get the following:
1/r 1
b
− t tα f a m1 − tα f b dt
Sa f α, m, r ≤
r
0
1/2 1 α t f a m1 − tα f b dt
t−
r
1/r
2.6
International Journal of Mathematics and Mathematical Sciences
5
r−1/r r−1
− t tα f a m1 − tα f b dt
r
1/2
1
r − 1 α t f a m1 − tα f b dt
t−
r
r−1/r
1/2 1 α
α
1
r − 1
≤
r − tt dt r − tt dt f a
0
1/2
1/2 1 1
r − 1
α
α
r − t1 − t dt r − t1 − t dt m f b
0
1/2
μ11 μ21 μ31 μ41 f a ν11 ν12 ν13 ν14 mf b,
2.7
which completes the proof.
Corollary 2.3. In Theorem 2.2, i if we choose α 1 and r 2, then we have the following:
mb
fa fmb
1
b
−
fxdx
mb − aSa f 1, m, 2 2
mb − a a
mb − a ≤
f a mf b ,
8
2.8
and ii if we choose α 1 and r 6, then we have the following
mb
1
a mb
1
b
fmb −
fxdx
mb − aSa f 1, m, 6 fa 4f
6
2
mb − a a
5
≤
mb − a f a mf b .
72
2.9
α
Theorem 2.4. Under the same notations in Theorem 2.2, if |f |q ∈ Km
a, b, for some α, m ∈
2
0, 1 , mb > a and q > 1 with 1/p 1/q 1, then, for any r ≥ 2, the following inequality holds:
b
Sa f α, m, r
q 1/q 4 q
q 1/q
1 1/p 1 q
1 4 ≤
μ1 f a ν1 m f b
μ1 f a ν1 m f b
2r 2
1/p q
q 1/q 3 q
q 1/q
1 r−2 2
2 2 3 μ1 f a ν1 m f b
μ1 f a ν1 m f b
.
8
r
2.10
6
International Journal of Mathematics and Mathematical Sciences
Proof. From Lemma 2.1 and using the properties of modulus, we have the following:
1/r 1
b
− t f ta m1 − tbdt
Sa f α, m, r ≤
r
0
1/2 1 f ta m1 − tbdt
t−
r
1/r
r−1/r r−1
− t f ta m1 − tbdt
r
1/2
1
r − 1 f ta m1 − tbdt.
t−
r
r−1/r
2.11
Using the power-mean integral inequality and α, m-convexity of |f |q for any t ∈
0, 1, we have the following
a
1/r 0
q
q
q
1
− t f ta m1 − tb dt ≤ μ11 f a ν11 mf b ,
r
2.12
b
1/2 q
q
q
1 fta m1 − tb dt ≤ μ21 f a ν12 mf b ,
r
2.13
q
q
q
r−1
− t fta m1 − tb dt ≤ μ31 f a ν13 mf b ,
r
2.14
q
q
q
r − 1 fta m1 − tb dt ≤ μ41 f a ν14 mf b .
t−
r
2.15
1/r
t−
c
r−1/r 1/2
d
1
r−1/r
By the similar way as the above inequalities a–d, we have the following:
a 1/r 0
q 1/q
1 1/p 1 q
1
1 f
f
− t f ta m1 − tb dt ≤
μ
ν
m
,
a
b
1
1
r
2r 2
2.16
International Journal of Mathematics and Mathematical Sciences
7
b 1/2 1/r
1/p q
1/q
1 r−2 2
1 μ21 f a ν12 mf b
f ta m1 − tb dt ≤
,
t−
r
8
r
2.17
c r−1/r 1/2
1/p q
q 1/q
1 r−2 2
r−1
μ31 f a ν13 mf b
− t f ta m1 − tb dt ≤
,
r
8
r
2.18
d 1
r−1/r
t−
q 1/q
1 1/p 4 q
r − 1 4 f ta m1 − tbdt ≤
f
f
μ
ν
m
.
a
b
1
1
r
2r 2
2.19
By 2.11 and 2.16–2.19 the assertion 2.10 holds.
Corollary 2.5. In Theorem 2.4, i if we choose α 1 and r 2, then we have that
mb
fa fmb
1
−
fxdx
2
mb − a a
q
q 1/q q
q 1/q 6−1/q
≤
,
5f a mf b
mb − a f a 5mf b
8
2.20
and ii if we choose α 1 and r 6, then we have that
mb
1
a mb
1
fxdx
fmb −
fa 4f
6
2
mb − a a
1/q
1
1
f aq 17mf bq 1/q
≤ mb − a
72
18
2/q
25
2
288
5
q
q 1/q q
q 1/q .
× 7f a 11mf b
11f a 7mf b
q
q 1/q 17f a mf b
2.21
8
International Journal of Mathematics and Mathematical Sciences
α
a, b, for some α, m ∈
Theorem 2.6. Under the same notations in Theorem 2.2, if |f |q ∈ Km
2
0, 1 , mb > a and q > 1 with 1/p 1/q 1, then, for any r ≥ 2, the following inequality holds:
b
Sa f α, m, r
q 1/q 4 q
q 1/q
1 1/p 1 q
1 4 ≤
μ2 f a ν2 m f b
μ2 f a ν2 m f b
r p1
1/p q
q 1/q 3 q
q 1/q
r − 2p1
2 2 3 μ2 f a ν2 m f b
μ2 f a ν2 m f b
,
2p1 r p1
2.22
where
μ12 1
r α1 α
1
,
μ22 r α1 − 2α1
,
2α1 r α1 α 1
μ32 2α1 r − 1α1 − r α1
,
2α1 r α1 α 1
μ42 r α1 − r − 1α1
,
r α1 α 1
ν21 1
− μ12 ,
r
2.23
r−2
μ22 ,
2r
r−2
ν23 μ32 ,
2r
1
ν24 − μ42 .
r
ν22 Proof. Suppose that q > 1. From Lemma 2.1, using the Hölder integral inequality, we get the
following:
b
Sa f α, m, r ≤
1/r 1/q
p 1/p 1/r
q
1
f tb m1 − ta dt
− t dt
r
0
0
1/2 1/q
1/p 1/2
q
1 p
f tb m1 − ta dt
dt
t−
r
1/r
1/r
r−1/r 1/q
p 1/p r−1/r
q
r−1
f tb m1 − ta dt
− t dt
r
1/2
1/2
1
1/q
1/p 1
q
r−1 p
f tb m1 − ta dt
dt
t−
r
r−1/r
r−1/r
International Journal of Mathematics and Mathematical Sciences
≤
1
1/p 1/r
9
1/q
f tb m1 − taq dt
r p1
0
1/q
1/p 1/2
q
1
r − 2 p1
f tb m1 − ta dt
r
2p1
1/r
1/q
1/p r−1/r
q
1
r − 2 p1
f tb m1 − ta dt
r
2p1
1/2
1/q
1
q
1 1/p
f tb m1 − ta dt
,
r p1
r−1/r
2.24
where we have used the fact that 1/2 < 1/p 11/p < 1.
Since |f |q ∈ Kαm a, b for some fixed α ∈ 0, 1 and m ∈ 0, 1, we have the followings:
1/r
0
1/2
f tb 1 − taq dt ≤ μ1 f aq ν1 mf bq ,
2
2
ftb 1 − taq dt ≤ μ2 f aq ν2 mf bq ,
2
1/r
r−1/r
1/2
1
r−1/r
2
f tb 1 − taq dt ≤ μ3 f aq ν3 mf bq ,
2
2.25
2
ftb 1 − taq dt ≤ μ4 f aq ν4 mf bq .
2
2
Hence, if we combine the inequalities in 2.24-2.25, we get the desired result.
Corollary 2.7. In Theorem 2.6, i if we choose α 1 and r 2, then we have that
mb
fa fmb
1
−
fxdx
2
mb − a a
2.26
11/q
q 1/q q
q 1/q q
1
≤
,
3 f a m f b
mb − a f a 3m f b
4
and ii if we choose α 1 and r 6, then we have
mb
1
a mb
1
fxdx
fmb −
fa 4f
6
2
mb − a a
12/q
1
f aq 11mf bq 1/q 11f aq mf bq 1/q
≤ mb − a
6
21/q q 1/q q
q 1/q q
1
f a 2m f b
,
2 f a m f b
3
2.27
where we have used the fact that 1/21/q < 1.
10
International Journal of Mathematics and Mathematical Sciences
3. Applications to Special Means
Now using the results of Section 2, we give some applications to the following special means
of positive real numbers a, b ∈ R with b ≥ a.
1 The arithmetic mean: Aa, b a b/2.
√
2 The geometric mean: Ga, b ab.
3 The logarithmic mean: La, b b − a/ln b − ln a for a /
b.
4 The harmonic mean: Ha, b 2ab/a b.
5 The power mean: Mr a, b ar br /21/r , r ≥ 1, a, b ∈ R.
6 The generalized logarithmic mean:
bn1 − an1
Ln a, b b − an 1
1/n
,
a/
b.
3.1
7 The identric mean:
⎧
⎪
⎪
⎨a 1/b−a
Ia, b 1 bb
⎪
⎪
⎩ e aa
ab
a/
b.
3.2
Proposition 3.1. For n ∈ −∞, 0 ∪ 1, ∞ \ {−1} and a, b ∈ 0, b∗ with b∗ > 0, we have the
following inequalities:
mb − a
n−1
a, m1/n−1 b .
a |Aan , mn bn − Lnn a, mb| ≤
|n|Mn−1
8
1
2
n
n n
n
n
b Aa , m b A a, mb − Ln a, mb
3
3
5
n−1
≤
a, m1/n−1 b .
mb − a|n|Mn−1
72
3.3
Proof. The assertions follow from Corollary 2.3 for fx xn .
Proposition 3.2. For a, b ∈ 0, b∗ , we have the following inequalities:
mb − a M22 m1/2 a, b ,
a H −1 a, mb − L−1 a, mb ≤
2
2
8a b
1 −1
5mb − a
2
2
1/2
m
a,
b
.
M
b H a, mb A−1 a, mb − L−1 a, mb ≤
2
3
3
72a2 b2
Proof. The assertions follow from Corollary 2.3 for fx 1/x.
3.4
International Journal of Mathematics and Mathematical Sciences
11
Proposition 3.3. For n ∈ −∞, 0 ∪ 1, ∞ \ {−1} and a, b ∈ 0, b∗ , we have the following
inequalities:
a |Aan , mn bn − Lnn a, mb|
1/q
1/q 3−1/q
1/q
n−1q
n−1q
1/q
n−1q
n−1q
≤
a
5a
,
, 5mb
A
, mb
mb − a|n| × A
8
1
2
b Aan , mn bn An a, mb − Lnn a, mb
3
3
1/q
1
1
≤
mb − a|n| × A1/q an−1q , 17mbn−1q A1/q 17an−1q , mbn−1q
72
9
1/q
25
2
mb − a|n|
288
25
× A1/q 7an−1q , 11mbn−1q A1/q 11an−1q , 7mbn−1q .
3.5
Proof. The assertions follow from Corollary 2.5 for fx xn .
Proposition 3.4. For n ∈ −∞, 0 ∪ 1, ∞ \ {−1} and a, b ∈ 0, b∗ , we have the following
inequalities:
1
a |Aan , mn bn − Lnn a, b| ≤
|n|mb − a
221/q
× A1/q an−1q , 3mbn−1q A1/q 3an−1q , mbn−1q ,
1
2 n
n
n n
n
b Aa , m b A a, mb − Ln a, mb
3
3
11/q 1/q
1
1
≤
|n|mb − a × A1/q an−1q , 11mbn−1q A1/q 11an−1q , mbn−1q
6
3
21/q 1
21/q |n|mb − a × A1/q an−1q , 2mbn−1q A1/q 2an−1q , mbn−1q .
3
3.6
Proof. The assertions follow from Corollary 2.7 for fx xn .
Proposition 3.5. For a, b ∈ 0, b∗ , we have the following inequalities:
Ia, b b − a
1
≤
a ln
aq 3bq 1/q 3aq bq 1/q ,
11/q
Ga, b
ab
4
12/q
b−a
Ia, b
1
q
q 1/q
q
q 1/q
≤
11b
b
11a
b ln 1/3
a
ab
6
G a, bA2/3 a, b 21/q 1
q
q 1/q
q
q 1/q
.
a 2b 2a b 3
Proof. The assertions follow from Corollary 2.7 for fx − ln x and m 1.
3.7
12
International Journal of Mathematics and Mathematical Sciences
Acknowledgment
The author is so indebted to the referee who read carefully through the paper very well and
mentioned many scientifically and expressional mistakes.
References
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m-convex and α, m-convex functions,” rgmia.org/papers/v13e/oss5.pdf.
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6 J. Park, “Generalizations of simpson-like type inequalities for differentiable m-convex and α, m convex mappings,” Far East Journal of Mathematical Sciences, vol. 55, no. 1, pp. 97–109, 2011.
7 M. Z. Sarikaya, E. Set, and M. E. Özdemir, “On new inequalities of Simpson’s type for functions whose
second derivatives absolute values are convex,” RGMIA Research Report Collection, vol. 13, no. 1, article
1, 2010.
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Inequalities, vol. 3, no. 3, pp. 305–313, 2009.
9 S. S. Dragomir, R. P. Agarwal, and P. Cerone, “On Simpson’s inequality and applications,” Journal of
Inequalities and Applications, vol. 5, no. 6, pp. 533–579, 2000.
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Optimization
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http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014